Nonlinear ordinary differential equations

(symmetries and singularities)

G.P.Flessas, P.G.L.Leach

University of the Aegean

Karlovassi 83 200, Greece

 

We provide a comprehensive review of the applications of the methods of symmetry analysis due to Lie and of singularity analysis due to Painlevé to nonlinear ordinary differential equations. In each case we present a brief development of the theoretical background to make this paper self-contained although there are many works cited for those interested to see the source material. A major emphasis in the symmetry analysis is the need to extend one's considerations past the traditional point and contact symmetries through generalized symmetries to the more recently elaborated nonlocal symmetries. The relationship between the two concepts of integrability, videlicet that of Lie and that of Painlevé, is examined not in the hope of establishing finality but with the intention to reveal the delicacy and complexity of the relationship between the two approaches to the integrability of nonlinear ordinary differential equations.

 

 

Giorgos Pavlos Flessas, Dr., Graduate of the University of Athens, Greece, and the Technical University, Wien; spent some ten years at the Department of Mathematics, University of Glasgow, before returning to Greece, firstly to the Mathematics Department of the University of Patras and secondly in 1989 as foundation Professor of Mathematics at the newly established University of the Aegean; active in the analysis of differential equations arising in Mathematical Physics, especially General Relativity and Cosmology.

Peter Gavin Lawrence Leach, Dr., Graduate of the University of Melbourne and La Trobe University in Australia and the University of Natal in South Africa; Professor of Mathematics of the University of Natal, Durban; formerly Professor of Applied Mathematics, University of the Witwatersrand, Johannesburg; active in the applications of symmetry and singularity analyses to differential equations arising in Mechanics, Quantum Mechanics, Cosmology, Mathematical Economics and Mathematical Biosciences; also interested in problems of molecular structure.